Reduced 1-cohomology and relative property (T) (Q415508)

Let \(G\) be a locally compact group. It is known that the Kazhdan's property (T) for a \(\sigma\)-compact \(G\) is equivalent to the vanishing of the first cohomology group \(H^1(G, \pi) = Z^1(G, \pi)/B^1(G, \pi)\) for every unitary representation \(\pi\), and for a compactly generated group \(G\), this is also equivalent to the vanishing of the first reduced cohomology group \({\overline H}^1(G, \pi) = Z^1(G, \pi)/{\overline B}^1(G, \pi)\) for every unitary representation \(\pi\). In this paper, the authors characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes. It is also shown that this is inequivalent to relative property (T) even for compactly generated group pairs.